volume 4, issue 4, article 79, 2003.
Received 15 July, 2002;
accepted 10 June, 2003.
Communicated by:N.E. Cho
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Journal of Inequalities in Pure and Applied Mathematics
PARTIAL SUMS OF FUNCTIONS OF BOUNDED TURNING
JAY M. JAHANGIRI AND K. FARAHMAND
Kent State University Burton, Ohio 44021-9500, USA.
EMail:jay@geauga.kent.edu University of Ulster,
Jordanstown, BT37 0QB, United Kingdom.
EMail:k.farahmand@ulster.ac.uk
c
2000Victoria University ISSN (electronic): 1443-5756 080-02
Partial Sums of Functions of Bounded Turning
Jay M. Jahangiri and K. Farahmand
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Abstract
We determine conditions under which the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning.
2000 Mathematics Subject Classification:Primary 30C45; Secondary 26D05.
Key words: Partial Sums, Bounded Turning, Libera Integral Operator.
Contents
1 Introduction. . . 3 2 Preliminary Lemmas. . . 5 3 Proof of the Main Theorem . . . 7
References
Partial Sums of Functions of Bounded Turning
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1. Introduction
LetAdenote the family of functionsf which are analytic in the open unit disk U ={z :|z|<1}and are normalized by
(1.1) f(z) =z+
∞
X
k=2
akzk, z ∈ U.
For0 ≤ α < 1,letB(α)denote the class of functionsf of the form (1.1) so that <(f0)> αinU. The functions inB(α)are called functions of bounded turning (c.f. [3, Vol. II]). By the Nashiro-Warschowski Theorem (see e.g. [3, Vol. I]) the functions inB(α)are univalent and also close-to-convex inU.
Forf of the form (1.1), the Libera integral operatorF is given by F(z) = 2
z Z z
0
f(ζ)dζ =z+
∞
X
k=2
2
k+ 1akzk.
Then-th partial sumsFn(z)of the Libera integral operatorF(z)are given by Fn(z) =z+
n
X
k=2
2
k+ 1akzk.
In [5] it was shown that iff ∈ Ais starlike of orderα, α = 0.294...,then so is the Libera integral operatorF.We also know that (see e.g. [1]), there are functions which are univalent or spiral-like in U so that their Libera integral operators are not univalent or spiral-like in U. Li and Owa [4] proved that if f ∈ A is univalent in U, then Fn(z) is starlike in|z| < 38. The number 38 is
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sharp. In this paper we make use of a result of Gasper [2] to provide a simple proof for the following theorem.
Theorem 1.1 (Main Theorem). If 14 ≤ α < 1 and f ∈ B(α), then Fn ∈ B 4α−13
.
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2. Preliminary Lemmas
To prove our Main Theorem, we shall need the following three lemmas. The first lemma is due to Gasper ([2, Theorem 1]) and the third lemma is a well- known and celebrated result (c.f. [3, Vol. I]) which can be derived from Her- glotz’s representation for positive real part functions.
Lemma 2.1. Letθ be a real number andmandkbe natural numbers. Then
(2.1) 1
3+
m
X
k=1
cos(kθ) k+ 2 ≥0.
Lemma 2.2. Forz∈ U we have
<
m
X
k=1
zk k+ 2
!
>−1 3.
Proof. For0≤r <1and for0≤ |θ| ≤πwritez =reiθ =r(cos(θ)+isin(θ)).
By DeMoivre’s law and the minimum principle for harmonic functions, we have
(2.2) <
m
X
k=1
zk k+ 2
!
=
m
X
k=1
rkcos(kθ) k+ 2 >
m
X
k=1
cos(kθ) k+ 2 .
Now by Abel’s lemma (c.f. Titchmarsh [6]) and condition (2.1) of Lemma2.1 we conclude that the right hand side of (2.2) is greater than or equal to −13 .
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Lemma 2.3. Let P(z)be analytic in U, P(0) = 1,and <(P(z)) > 12 in U. For functionsQanalytic inU the convolution functionP∗Qtakes values in the convex hull of the image onU underQ.
The operator “∗” stands for the Hadamard product or convolution of two power seriesf(z) = P∞
k=1akzkandg(z) = P∞
k=1bkzkdenoted by(f∗g)(z) = P∞
k=1akbkzk.
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3. Proof of the Main Theorem
Letf be of the form (1.1) and belong toB(α)for 14 ≤α <1.Since<(f0(z))>
αwe have
(3.1) < 1 + 1
2(1−α)
∞
X
k=2
kakzk−1
!
> 1 2.
Applying the convolution properties of power series toFn0(z)we may write Fn0(z)
(3.2)
= 1 +
n
X
k=2
2k
k+ 1akzk−1
= 1 + 1
2(1−α)
∞
X
k=2
kakzk−1
!
∗ 1 + (1−α)
n
X
k=2
4 k+ 1zk−1
!
=P(z)∗Q(z).
From Lemma2.2form =n−1we obtain
(3.3) <
n
X
k=2
zk−1 k+ 1
!
>−1 3.
Applying a simple algebra to the above inequality (3.3) andQ(z)in (3.2) yields
<(Q(z)) =< 1 + (1−α)
n
X
k=2
4 k+ 1zk−1
!
> 4α−1 3 .
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On the other hand, the power seriesP(z)in (3.2) in conjunction with the con- dition (3.1) yields<(P(z))> 12.Therefore, by Lemma2.3, <(Fn0(z))> 4α−13 . This concludes the Main Theorem.
Remark 3.1. The Main Theorem also holds forα < 14.We also note thatB(α) forα <0is no longer a bounded turning family.
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References
[1] D.M. CAMPBELLANDV. SINGH, Valence properties of the solution of a differential equation, Pacific J. Math., 84 (1979), 29–33.
[2] G. GASPER, Nonnegative sums of cosines, ultraspherical and Jacobi poly- nomials, J. Math. Anal. Appl., 26 (1969), 60–68.
[3] A.W. GOODMAN, Univalent Functions, Vols. I & II, Mariner Pub. Co., Tampa, FL., 1983.
[4] J.L. LI AND S. OWA, On partial sums of the Libera integral operator, J.
Math. Anal. Appl., 213 (1997), 444–454.
[5] P.T. MOCANU, M.O. READE AND D. RIPEANU, The order of starlike- ness of a Libera integral operator, Mathematica (Cluj), 19 (1977), 67–73.
[6] E.C. TITCHMARSH, The Theory of Functions, 2nd Ed., Oxford University Press, 1976.